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Statistical Morphology is concerned with the statistical characterization of the four morphological operations — dilation, erosion, opening and closing. By statistical characterization of a morphological operator we mean the statistical characterization of the output in terms of the statistical characteristics of the input. Characterization of operators allows us to predict the characteristics of the output of an algorithm composed of a sequence of morphological operations in terms of the statistical characteristics of the input and the sequence of morphological operators used. Furthermore, such statistical analyses of morphological algorithms is necessary for evaluating the algorithm's performance. In this paper we describe what we have learned about one way to characterize the dilation and opening morphological operators in a one dimensional setting. That is, the input to each of these operators is assumed to be binary one-dimensional. The input is modeled as a union of randomly translated discrete lines of a fixed length. The line segments can overlap and result in line segments of various lengths. Thus the final output appears as an unordered pattern of lines and gaps of various lengths. This input is characterized by giving its line and gap length distribution and the distribution of the number of line and gap segments of various lengths. The characterization of a morphological operator, therefore, entails a similar characterization of the output. There has been a recent interest in the area of statistical morphology and some results have been published in the literature. Morales and Acharya {MA92} analyzed the statistical characteristics of a morphological opening on grayscale signals perturbed by Gaussian noise. Stevenson and Arce {SA92] studied the effects of opening for a class of structuring elements. Atola, Koskinen and Neuvo [ALN93] studied the output distributions of one dimensional grayscale filtering. Costa and Haralick {CH92} came up with an empirical description of the output graylevel distributions of morphologically opened signals. Dougherty and Loce [DL93] used libraries of structuring elements to restore corrupted signals in the case when a noise model is available. In the following section we set up the notation and definitions used in this paper. In section 3 we give a formal statement of the random process used to generate random sequences. In section 4 we give a maximum likelihood algorithm for estimating the model parameters. The four morphological operators are characterized in section 5.
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